# Number Theory: Unique Numbers

#### Islam Hassan

Does anyone know of a reference work that lists natural numbers with unique properties? Like 26, for example, being the only natural number sandwiched between a square (25) and a cube (27). Does such a reference book exist?

IH

#### micromass

Wikipedia has such information: http://en.wikipedia.org/wiki/38_(number [Broken])

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#### Islam Hassan

Thanx Micro, I was aware of certain Wikipedia articles; what I specifically am looking for though, is a systematic reference work of all know unique numbers. I could not find something resembling this on the net...

#### mfb

Mentor
It depends a lot on the things you consider as unique properties. Every number has unique properties, but most of them are boring ("is the only number x where x-23 and x-24 are primes" is another one for 26). Random collections are the best things you can find.

#### Islam Hassan

Hmm...is that a trivial uniqueness quality that you just mentioned for 26? Doesn't seem so to me but then I am the layman here...

Can one somehow 'define' mathematical triviality for such unique qualities I wonder...

IH

#### mfb

Mentor
It is trivial in the way that "x-23 prime and x-24 prime" requires two primes with a difference of just 1, and 2 and 3 are the only primes that satisfy this.
You can set this up for every integer.

#### Islam Hassan

Yes, if course...silly me...

#### Islam Hassan

Thanx Curious, exactly the type of thing I was looking for, thanx a million...I kept repeating "unique" in all my Google searches, so there you go...a little variety is always good...

IH

#### mfb

Mentor
Note that not all those entries are unique, and some of them just reflect our limited knowledge. And some are... pointless.

"151 is a palindromic prime." - true, but there are 7 smaller palindromic primes and probably infinitely more larger ones.
"146 = 222 in base 8." - so what?

#### Islam Hassan

Thanx for the clarification mob...funny I would have thought that a compendium of numbers with unique characteristics would be a given in number theory...quite surprised that it's so difficult to find...

IH

#### kjhskj75

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